True exhaust gas temperatures and, therefore, housing temperatures inside of the turbo via thermal conduction are likely to approach 1600 degrees at 6,000 rpm when the turbo is under full load, as in 4th through 6th gear. That is normal. Once the cast iron housing gets heated all the way through, temps on the surface can be close to those inside, even with a water jacket built into the housing.
There is heat shielding around the turbo and the downpipe for a very good reason.
600 degrees is more like the temp on cooled down turbo with the engine still running.
I run a catless exhaust system. The outside of my stainless exhaust tip is discolored in a bluish, yellowish, orangish range from heat (like color case hardening on firearms) even that far back. See attached photo. Some guys pay extra for fake "burnt exhaust" tips. Run the extra heat of a catless system with a single wall exhaust tip, do a little "spirited driving" in the upper gears and get the real thing, rather than just the "look."
Turbocharged engines under load and pushing the outer limits of their thermal efficiency range produce a LOT of heat.